Capillary pressure data are obtained on small core samples that represent an extremely small part of the reservoir and, therefore, it is necessary to combine all capillary data to classify a particular reservoir. The fact that the capillary pressure-saturation curves of nearly all naturally porous materials have many features in common has led to attempts to devise some general equation describing all such curves. Leverett (1941) approached the problem from the standpoint of dimensional analysis. At first, Leverett made an attempt to convert all capillary pressure data to a universal curve. But, a universal capillary pressure curve does not exist because the rock properties affecting capillary pressures in reservoir have extreme variation with lithology (rock type). Realizing that capillary pressure should depend on the porosity, interfacial tension, and mean pore radius, Leverett defined the dimensionless function of saturation, which he called the J-function, as
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√ (2-116)
Where, c is constant.
In fact, The Leverett J-function is an attempt at extrapolating capillary pressure data for a given rock to rocks that are similar but with differing permeability, porosity and wetting properties. It assumes that the porous rock can be modelled as a bundle of non-connecting capillary tubes, where the factor √ is a characteristic length of the capillaries' radii and Leverett interpreted √ as being proportional to the square of a mean pore radius.
For the same formation, however, this dimensionless capillary-pressure function serves quite well in many cases to remove discrepancies in the pc versus Sw curves and reduce them to a common curve. Some authors altered eq. (2-116) by including the :
√ (2-117)
Field Scale example: Browns (1951) considered the j-function (eq. (2-116)) as a correlating device for capillary pressure data of samples from Edwards formation in the Jourdanton field. () shows the correlation obtained for all samples available from the field. There is considerable dispersion of data points, although the trend of correlation is good. If core samples are divided to two different groups (limestone and dolomites) according to their texture, the correlation will be improved. () shows the correlations for the limestone and dolomite samples respectively. The dolomite samples show a good correlation while the limestone sample data shows a scattering of data in the range of lower water saturation. If the limestone samples are subdivided to micro-granular and coarse-grained samples, the correlation for each of them will be improved (). Scattering of data for the coarse-grained limestone samples are greater than other groups of samples. This is to be expected , as the coarse-grained limestone contains solution cavities, vugs and channels which are not capillary in size, hence deviation from trends established in capillary-pressure data.
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(c) (d)
(a). J curve for all cores
(b). J curve for Limestone cores (c). J curve for dolomite cores
(d). J curve for micro-granular limestone cores (e). J curve for coarse-grained limestone cores
(e)
Figure 2-73- J-function correlation of capillary pressure data in Edwards Jourdanton field.
Example 2-7
A laboratory capillary pressure test was conducted on a core sample taken from the Nameless Field. The core has a porosity and permeability of 16% and 80 md, respectively. The capillary pressure-saturation data are given as follows:
Sw Pc (psi) 1.0 0.50 0.8 0.60 0.6 0.75 0.4 1.05 0.2 1.75
The interfacial tension is measured at 50 dynes/cm. Further reservoir engineering analysis indicated that the reservoir is better described at a porosity value of 19% and an absolute permeability of 120 md. Generate the capillary pressure data for the reservoir.
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Solution
Calculate the J-function using the measured capillary pressure data and eq. (2-116). √ √ Sw Pc (psi) J(Sw) 1.0 0.50 0.048 0.8 0.60 0.058 0.6 0.75 0.073 0.4 1.05 0.102 0.2 1.75 0.169
At the next step, Using the new porosity and permeability values, solve eq. (2-116) for the capillary pressure Pc. √⁄ √ Sw J(Sw) Pc (psi) 1.0 0.048 0.441 0.8 0.058 0.533 0.6 0.073 0.671 0.4 0.102 0.938 0.2 0.169 1.553
RELATIVE PERMEABILITY
The absolute permeability is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids. The absolute permeability is referred to the rock condition where single phase saturation is considered in other words the rock is fully saturated with one homogeneous fluid. Fluid flow in reservoirs typically involves more than one fluid, which means that the general ability of one fluid to flow is affected by the presence of other fluids in the reservoir. In order to develop estimates of fluid behavior in reservoirs, this phenomenon needs to be quantified in some way. The concept of describing this multiphase flow in reservoirs is known as relative permeability, which is defined as the ratio of the effective permeability of a fluid to the absolute permeability of the rock. The effective permeability is a relative measure of conductance of the porous medium for one fluid phase in the presence of other fluid phases.
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Where,
kro = relative permeability to oil krg = relative permeability to gas krw = relative permeability to water k = absolute permeability
ko = effective permeability to oil for a given oil saturation kg = effective permeability to gas for a given gas saturation kw = effective permeability to water for a given water saturation
In the definition of effective permeability each fluid phase is considered to be completely independent of the other fluids in the flow network. The fluids are considered immiscible, so that Darcy’s law can be applied to each individually:
( ) (2-118) ( ) (2-119) ( ) (2-120) Where, s is direction.
Since the fluid is not completely saturating the rock, other factors come into play other than the pore size distribution. According to experimental evidence, effective permeability is a function of dominant fluid saturation, saturation history, wettability characteristic and pore geometry. So to determine the relative permeability for one phase it is necessary to specify the fluid saturation. In contrast to absolute permeability, relative permeability for each phase is not a unique value and there is a special relative permeability value for each saturation amount.
Since the effective permeabilities may range from zero to k, the relative permeabilities may have any value between zero and one. It should be pointed out that when three phases are present the sum of the relative permeabilities (kro + krg + krw) is both variable and always less than or equal to unity.